arXiv subjects

Gunhee Cho

Publications and source records attributed to Gunhee Cho.

At least 19 recordsLinked to original sources

arXiv2026

How the $p$-Bergman metric approaches the Carathéodory metric

We study the rate at which the $p$-Bergman metric approaches the Carathéodory metric on convex domains with measures comparable to $δ^αdV$, $α>-1$. On smooth convex domains of finite type the logarithmic coefficient is bounded above by the harmonic-measure average of the tangential multitype dimension plus $1+α$, with equality when almost every endpoint of the extremal disc is coercive. For a class of convex domains in $\mathbb{C}^2$ with $C^4$ boundary and rank-one real tangential contact along an axis disc, a winding number distinguishes the coefficient $2+α$ from an upper bound $7/4+α$; quartic contact makes the latter bound exact. The exact value holds along the entire axis of explicit real-algebraic strongly pseudoconvex domains, including perturbations without the weighted circle action, although their pointwise boundary models agree up to unitary equivalence with those of a domain having coefficient $2+α$. A $C^2$ example has negative winding but coefficient $2+α$, with a $-\frac12\log\log p$ correction, so boundary regularity is essential to the gap theorem.

math.CV↗

arXiv2026

FATE-5: Faithful Alignment of Trait Expression for Continuous Big Five Personality Control

Controlling personality in large language models is important for building conversational agents that can be customized to different users and interaction settings. However, existing personality control methods often face a tradeoff between strong trait expression and natural dialogue generation. We introduce FATE-5, Faithful Alignment of Trait Expression for the Big Five, a two-stage framework that separates personality encoding from naturalness alignment. FATE-5 first learns continuous soft prompt representations of Big Five profiles using dialogue supervision and an auxiliary survey objective. It then improves dialogue naturalness through preference alignment based on both dialogue quality and personality agreement. Across three language model backbones, FATE-5 maintains strong trait expression while achieving substantially higher dialogue naturalness than other trait-strong methods. On held-out questionnaires, it achieves monotonicity correlations of 0.97-0.99 and recovery RMSE of 8.9-12.4, substantially improving over the evaluated baselines. These results demonstrate effective continuous personality control while maintaining natural conversational behavior.

cs.CL↗

arXiv2026

Personality Latents: Conditioning Frozen Language Models on Individuals from Their Own Writing

Controlling the personality of a language model usually relies on written profiles or named trait categories, so it can only express what an author writes down or what a taxonomy names. We instead propose Personality Latents (PL), which learn a latent for each person from their own writing, with no personality labels and no authored profile, and inject it into a frozen language model as a soft prefix. Following generative latent optimization, each training person receives a free latent that is optimized through the frozen model itself, while an encoder learns to predict these latents so that a new person needs only one forward pass, with optional refinement at test time. We evaluate on held-out people by having the conditioned model answer neutral and interview questions and reading personality from its answers with scorers trained on real human writing. On the MBTI axes PL reaches an AUC of , ahead of persona embedding and prompt-based baselines under identical settings, and the ranking holds when the answers are scored in three different embedding spaces. A latent learned with Qwen2.5-3B-Instruct also transfers personality to Llama-3.1-8B-Instruct through a new adapter alone.

cs.CL↗

arXiv2026

Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information

A Brownian particle observed at time $t$ carries information about where it started. On a closed surface this makes the surface itself a statistical model, whose Fisher information pulls back to a heat-smoothed metric $h_t=2tJ_t$; its volume $V_t$ is a normalized Jeffreys volume, and $\log(V_t/2t)$ is the model-dependent constant in Rissanen's minimum-description-length complexity. We ask whether maximizing this statistical size among metrics of fixed area in a conformal class recovers the uniformizing metric. First, $9(A-2πχt-V_t)/t^2$ converges to the curvature energy $\int K^2\,dA$ (the Calabi energy up to a topological constant), uniformly on classes of bounded geometry; hence, within a fixed-area conformal class of bounded geometry containing the uniformizer, approximate maximizers of $V_t$ converge to the constant-curvature metric (on the sphere, to the round metrics). Second, near every flat torus we prove a sharp finite-scale version: for small perturbations whose oscillations are resolved by the diffusion, $t|k|^2\leθ<τ_*$, the flat-subtracted defect is comparable to $\|Δf\|_2^2$, uniformly in the frequency cutoff and over noncollapsing lattices. The explicit threshold $τ_*=4.408\ldots$ is optimal, and it is exactly the observation time at which the information about one curvature mode is maximal. Third, the flat structure is seen only at an exponential scale: the volume loss of a flat torus measures the ambiguity of winding and is comparable to $t^{-1/2}e^{-\operatorname{sys}^2/(16t)}$, which selects the hexagonal lattice as $t\to0$ but not at all times.

math.DG↗

arXiv2026

Fisher Completions and Statistical Regularization of Wrapped Gaussian Families

We determine the intrinsic Fisher completion of the full-covariance wrapped Gaussian family on a fixed torus and relate its geometry to uniform estimation. The completion is homeomorphic to the total-variation closure. Every added law is, in suitable integer torus coordinates, a product of a uniform law and a lower-dimensional wrapped Gaussian. All these densities are positive and real analytic, but in dimension at least two no added point has a compact neighborhood. Small balls remain contractible, and explicit minimizing geodesics pass through added points. In dimension at least two, every fixed Fisher neighborhood of an added point has a positive minimax risk at every sample size. For subfamilies with a common positive lower bound on the variance in every nonzero integer direction, we prove that compactness of the completed closure is equivalent to uniform consistency in either $L^1$ or completed-Fisher loss. Equivalently, the Fourier moments must decay uniformly over the subfamily. A fixed covariance floor guarantees this condition without an upper covariance bound and gives $L^1$ risk at most $C_{\varepsilon,d}n^{-1/2}\{\log(n+1)\}^{d/4}$. A common-phase example illustrates how partially uniform limits can retain relative-phase information while remaining uniformly estimable.

math.ST↗

arXiv2026

Log-Concavity and Fundamental Gaps on Complex Projective Domains

We prove quantitative log-concavity and fundamental-gap estimates for Dirichlet Schrödinger operators on complex projective domains. With holomorphic sectional curvature four, the boundary condition $\mathrm{II}\ge\sqrt{44/5}\,g$ suffices in every complex dimension under $\operatorname{Hess} V\ge2|\nabla V|g$ and $\operatorname{osc}V\le4n$. It yields $\operatorname{Hess}(-\logϕ_1)>g$ and $λ_2-λ_1>\max\{n+2+\sqrt{n^2+5n+11},\,2n+14/3\}$. The main difficulty is to control the distinguished Hopf direction together with a transverse Hessian block whose dimension grows with $n$. We extend modified-Hessian first-contact arguments by retaining the Hopf term and using longitudinal and transverse rotational tests. A trace minimization reduces all gradient orientations to three extreme directions. A dimension-scaled logarithmic barrier then compares their contact costs with a single limiting profile. A radial spectral estimate supplies the required quadratic-in-dimension spectral threshold from a dimension-independent boundary curvature bound. For $n\ge3$ these are, to our knowledge, the first log-concavity estimates for domains in $\mathbb{CP}^n$ without symmetry assumptions. We also prove a sharp diameter bound and Hessian levels linear in $n$ under larger dimension-free curvature bounds.

math.AP↗

arXiv2026

Scaling Personality Control in LLMs with Big Five Scaler Prompts

We present Big5-Scaler, a prompt-based framework for conditioning large language models (LLMs) with controllable Big Five personality traits. By embedding numeric trait values into natural language prompts, our method enables fine-grained personality control without additional training. We evaluate Big5-Scaler across trait expression, dialogue generation, and human trait imitation tasks. Results show that it induces consistent and distinguishable personality traits across models, with performance varying by prompt type and scale. Our analysis highlights the effectiveness of concise prompts and lower trait intensities, providing a efficient approach for building personality-aware dialogue agents.

cs.CL↗

arXiv2026

Complex Analysis and Riemann Surfaces: A Graduate Path to Algebraic Geometry

This book develops a computation-driven route from classical complex analysis to compact Riemann surfaces and algebraic geometry. It begins with holomorphic and meromorphic functions, Cauchy theory, power and Laurent series, residues, contour integration, branch cuts, harmonic functions, conformal mapping, invariant metrics, the Riemann mapping theorem, elliptic functions, and analytic continuation. Differential forms and Stokes' theorem lead to de Rham cohomology, curvature, Gauss-Bonnet, and Hodge theory. Coordinate calculations relate first fundamental forms to Hermitian and Kahler metrics and reduce curvature under conformal change to Poisson and Liouville equations; the sphere and torus are worked out explicitly. The Hodge-Weyl theorem is developed through weak formulations, elliptic regularity, harmonic representatives, top-degree cohomology, and the one-dimensional Calabi-Yau equation. The cohomological chapters introduce complex manifolds, line bundles, divisors, the Picard group, sheaves, Cech cohomology, derived functors, fine resolutions, de Rham and Dolbeault cohomology, the exponential sequence, canonical bundles, Chern classes, Stein Riemann surfaces, and the Mittag-Leffler problem. Branched covers and gluing remain recurring models. Duality and several proofs of Riemann-Roch lead to surface Riemann-Roch and intersection theory. The final parts treat Jacobians, Abel-Jacobi theory, algebraic curves, local intersection multiplicity, theta and Weierstrass functions, Mellin transforms and the zeta functional equation, Galois coverings, modular curves, and cohomological representations. Definitions and global theorems are paired with calculations, examples, and exercises for advanced undergraduates and beginning graduate students.

math.CV↗

arXiv2026

Completing or Refusing Low-Dimensional Records of Structured Quantum Circuits: Measurement Loss, Compression Loss, and Hardware Drift

Structured quantum circuits are often represented by low-dimensional records such as occupation numbers or constraint feasibility rather than by full outcome distributions. A record loses control-dependent information when it merges outcomes whose probabilities respond differently to circuit parameters. We assess this loss without assuming a parametric hardware-noise model. Quantum Fisher information bounds premeasurement sensitivity; Fisher metrics of measurements and recorded statistics describe device-attainable sensitivity. For finite control changes, a Hellinger residual measures the response removed by a record, while its local quadratic term is the conditional covariance of the full-outcome score. Simultaneous confidence bounds support approval, refusal, or deferral. We prove a finite-library completion theorem: executable augmentations terminate either with a record preserving every declared response or with proof that no library augmentation removes the loss. For an analytic control germ, integral closures characterize preservation along every analytic control arc, and finitely many Rees valuations detect failure. This connects the state--measurement--record chain to an executable completion-or-refusal procedure, an attainable-Fisher kernel criterion, and finite-sample decisions. A three-qubit calculation separates measurement loss from record loss. Prospectively fixed IBM Kingston and Marrakesh experiments test decisions in four-qubit fixed-particle-number and GHZ families, including negative controls. In two-epoch IQM Garnet data, median record-level drift is 0.0797 times full-distribution drift, but a significant residual remains in 35 of 36 settings. On these circuits, the rules detect loss and approve records only within stated tolerances; they do not establish universal compression performance or device quantum Fisher information.

quant-ph↗

arXiv2026

Propagation of Chaos on Riemannian Manifolds

We extend reflection coupling for uniform-in-time propagation of chaos from Euclidean space to complete Riemannian manifolds. Intrinsic reflection along minimizing geodesics replaces the Euclidean difference process, and the associated endpoint-index term quantifies the competition between curvature, confinement, and interaction. Under an endpoint-index bound and a radial Laplacian bound, we establish global well-posedness, uniform moment estimates, and uniform-in-time propagation of chaos with rate $N^{-1/2}$ in a modified Wasserstein distance. We further obtain exponential contraction of the nonlinear McKean--Vlasov semigroup, uniqueness of its invariant probability measure among probability measures with finite second moment, and exponential convergence to equilibrium. Although a Ricci lower bound is an important special case, our framework also includes a curvature-spike class with $\inf_M\operatorname{Ric}=-\infty$.

math.PR↗

arXiv2026

Certified Quotient Calibration with Weighted-Projective Orbits and Finite-Shot Guarantees

Periodic controls can make a quantum-calibration scan redundant, but compilation, noise, and measurement can invalidate the symmetry behind that reduction. We introduce Certified Quotient Calibration (CQC), which audits whether circuit amplitudes have a common positive grading and define weighted-projective-space (WPS) control orbits. From pilot counts it bounds the largest proposed within-orbit and smallest audited between-orbit squared Hellinger distances by U_in and L_out. CQC accepts only when these bounds meet a task tolerance, preserve audited alternatives, and imply a net saving after certification cost; otherwise it rejects or defers the reduction. The theorem bounds accepted orbit substitutions and [0,1]-valued calibration objectives on one confidence event. Its cost corollary authorizes a representative-only scan only when its certification-inclusive cost is below that of the full grid. Weighted-projective descent, Hellinger testing rates, and multinomial concentration are standard; the contribution is their combination into a finite-shot approve--reject--defer guarantee for task error and net calibration cost. Gate-level experiments on IQM Garnet and IBM Marrakesh, Kingston, and Fez exercise all three decisions in compiled one-, two-, and three-qubit families. In a preregistered two-qubit Bell-phase experiment on IBM Kingston, the pilot audit found U_in=0.01922<L_out=0.09030, and quotient and full scans selected the same correction orbit for both injected phase offsets. Fresh held-out counts met the prespecified 0.03 noninferiority margin. Including 18,432 pilot shots, the quotient procedure used 36,864 rather than 55,296 total shots, a 33.3% reduction. These results support the CQC decision path for the tested families, but not a native cubic pulse law, a device-independent WPS orbit, or a universal saving rate.

quant-ph↗

arXiv2026

A Tutorial on Bregman Projection in Statistics

A single geometric operation -- projecting a reference onto a constrained family under a Bregman divergence -- underlies a striking range of statistical methods. This tutorial develops the operation first as pure convex geometry, with no statistics attached. A strictly convex generator $G$ and its conjugate $F$ furnish two coordinate systems, a projection theorem with existence and uniqueness, and a Pythagorean {theorem}; the Pythagorean theorem itself produces {two} dual projections -- the information (e-) projection onto moment-constrained families and the moment (m-) projection onto exponential families -- exchanged by the conjugacy $G\leftrightarrow F$, so a single theorem governs both. Part~II reads off the statistics. The generalized linear model is treated in detail as the concrete carrier of the two projections: {under the canonical link,} the score equation is exactly the Pythagorean orthogonality, and the fit is simultaneously an e-projection in the natural coordinate and an m-projection in the mean coordinate. Maximum entropy, survey calibration, over-identified moment models, the EM algorithm, variational inference, autoencoders, and expectation propagation then fall into place as instances of the same construction -- exactly where the underlying families are flat, and as controlled approximations or neighboring-divergence analogies where they are not. The mathematics of Part~I is self-contained; the statistical sections presume only familiarity with the methods being unified.

math.ST↗

arXiv2026

Statistical Bergman geometry

This paper explores the Bergman geometry of bounded domains $Ω$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $Φ: Ω\rightarrow \mathcal{P}(Ω)$, where $\mathcal{P}(Ω)$ denotes a space of probability measures on $Ω$. A result by J. Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian pseudo-metric in information geometry, via $Φ$ coincides with the Bergman metric of $Ω$. Building on this idea, we consider $Ω$ as a statistical model and present several interesting results within this framework. First, we derive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: Ω_1 \rightarrow Ω_2$, we prove that if the induced measure push-forward $κ: \mathcal{P}(Ω_1) \rightarrow \mathcal{P}(Ω_2)$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish the consistency and the central limit theorem of the Fréchet sample mean for Calabi's diastasis function.

math.CV↗

arXiv2026

Geometry- and Topology-Informed Quantum Computing: From States to Real-Time Control with FPGA Prototypes

This book gives a geometry-first, hardware-aware route through quantum-information workflows, with one goal: connect states, circuits, and measurement to deterministic classical pipelines that make hybrid quantum systems run. Part 1 develops the backbone (essential linear algebra, the Bloch-sphere viewpoint, differential-geometric intuition, and quantum Fisher information geometry) so evolution can be read as motion on curved spaces and measurement as statistics. Part 2 reframes circuits as dataflow graphs: measurement outcomes are parsed, aggregated, and reduced to small linear-algebra updates that schedule the next pulses, highlighting why low-latency, low-jitter streaming matters. Part 3 treats multi-qubit structure and entanglement as geometry and computation, including teleportation, superdense coding, entanglement detection, and Shor's algorithm via quantum phase estimation. Part 4 focuses on topological error correction and real-time decoding (Track A): stabilizer codes, surface-code decoding as "topology -> graph -> algorithm", and Union-Find decoders down to microarchitectural/RTL constraints, with verification, fault injection, and host/control-stack integration under product metrics (bounded latency, p99 tails, fail-closed policies, observability). Optional Track C covers quantum cryptography and streaming post-processing (BB84/E91, QBER/abort rules, privacy amplification, and zero-knowledge/post-quantum themes), emphasizing FSMs, counters, and hash pipelines. Appendices provide visualization-driven iCEstick labs (switch-to-bit conditioning, fixed-point phase arithmetic, FSM sequencing, minimal control ISAs), bridging principles to implementable systems.

quant-ph↗

arXiv2026

Hyperbolic Embeddings in Toric Geometry: Effectivity and Deformation Stability

We study the deformation behavior of Kobayashi hyperbolic embeddings for complements of divisors in projective toric varieties. In the toric setting, entire curves in divisor complements propagate along algebraic subtori, allowing hyperbolicity questions to be translated into combinatorial conditions on lattice-point configurations of Newton polytopes. Building on a theorem of Tiba, which guarantees hyperbolic embedding for a general divisor under suitable facewise lattice conditions, we develop an effective refinement of his argument. We construct an explicit Zariski closed exceptional locus in the coefficient parameter space, characterized by the presence of translated subtori in the support or complement of the divisor. This description makes the exceptional set amenable to explicit computation. Using this effectivity, we prove a deformation stability result: along any algebraic one--parameter family of divisors whose initial member avoids the exceptional locus, hyperbolic embedding fails for at most finitely many parameters. Under strengthened lattice-point hypotheses, we further exhibit distinguished one--parameter families for which hyperbolic embedding persists without any exceptional parameters. We also analyze deformations arising from diagonal torus reparametrizations, showing that the number of exceptional parameters in such similarity families is controlled by a simple multiplicative rank invariant of the scaling vector. Finally, we illustrate the theory through explicit examples in projective spaces and Hirzebruch surfaces, including a benchmark computation in $\mathbb{P}^3$ where the exceptional locus contains a hypersurface component of degree $126$.

math.AG↗

arXiv2025

Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation

We develop a unified geometric framework for quantum circuit compilation based on quantized orbifold phases and their diagrammatic semantics. Physical qubit platforms impose heterogeneous phase resolutions, anisotropic Bloch-ball contractions, and hardware-dependent $2π$ winding behavior. We show that these effects admit a natural description on the weighted projective line $\mathbb{P}(a,b)$, whose orbifold points encode discrete phase grids and whose monodromy captures winding accumulation under realistic noise channels. Building on this geometry, we introduce the WPL--ZX calculus, an extension of the standard ZX formalism in which each spider carries a weight--phase--winding triple $(a,α,k)$. We prove soundness of LCM-based fusion and normalization rules, derive curvature predictors for phase-grid compatibility, and present the Weighted ZX Circuit Compression (WZCC) algorithm, which performs geometry-aware optimization on heterogeneous phase lattices. To connect circuit-level structure with fault-tolerant architectures, we introduce Monodromy-Aware Surface-Code Decoding (MASD), a winding-regularized modification of minimum-weight matching on syndrome graphs. MASD incorporates orbifold-weighted edge costs, producing monotone decoder-risk metrics and improved robustness across phase-quantized noise models. All results are validated through symbolic and numerical simulations, demonstrating that quantized orbifold geometry provides a coherent and hardware-relevant extension of diagrammatic quantum compilation.

quant-ph↗

arXiv2025

Support-Projected Petz Monotone Geometry of Pure Two-Qubit Families: Universal Three-Channel Decomposition and Non-Reduction of Curvature Invariants

We develop a support-projected Petz monotone geometry for pure two-qubit families, obtained by pulling back arbitrary Petz monotone quantum metrics to circuit-defined submanifolds and projecting onto the active spectral support of the associated quantum Fisher information tensor. This framework strictly generalizes the symmetric logarithmic derivative (SLD/Bures) case and includes, as special examples, the Wigner--Yanase and Bogoliubov--Kubo--Mori metrics among many others.

quant-ph↗

arXiv2025

Structural Invariance of Green--Griffiths--Demailly Thresholds on Compact Complex Orbifolds

We prove that the Green--Griffiths--Demailly (GGD) hyperbolicity thresholds are structurally invariant. In other words, the minimal jet order and asymptotic growth rate at which invariant jet differentials appear remain unchanged when passing from a compact complex manifold to any compact smooth analytic Deligne--Mumford stack (orbifold) with the same coarse Kähler class. We establish an orbifold Riemann--Roch formula showing that only the identity sector contributes to the leading $m^n$ term of the Euler characteristic $χ$, while all twisted sectors contribute only $O(m^{n-1})$. Together with curvature--positivity properties of the Demailly--Semple tower, this implies that the existence range of invariant jet differentials depends solely on the coarse Kähler class--hence orbifold compactification or rigidification does not alter the GGD threshold or the hyperbolicity locus.

math.AG↗